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Nondestructive complex permittivity and permeability extraction using a two-layer dual-waveguide probe measurement geometry.

1. INTRODUCTION

Waveguide probes, whether they be coaxial, rectangular, or circular, have numerous applications including nondestructive evaluation of solids and liquids [1-14], surface and subsurface crack detection [15-19], and even in vivo characterization of biological tissues [20,21]. The majority of the published waveguide probe research deals with single probe geometries. The single probe geometry is well suited for the characterization of dielectric materials, e.g., biological specimens, plastics, ceramics, and most liquids; however, it suffers when one wants to fully characterize a magnetic material, such as a radio-frequency magnetic shielding material. This limitation prompted the development of techniques such as two-thickness method [14,22-24], two-layer method [10,11,13,24,25], frequency-varying method [26], and the short/free-space method [10,11,13] to measure both permittivity [[epsilon].sub.r] and permeability [[mu].sub.r] using a single probe. However, these methods are not always applicable to a given measurement and may be numerically unstable [25].

Methods capable of obtaining both reflection [S.sub.11] and transmission [S.sub.21] coefficients simultaneously allow complex [[epsilon].sub.r] and [[mu].sub.r] values to be determined unambiguously at every data point due to the independence of [S.sub.11] and [S.sub.21]. Two measurement geometries have been published which exploit this technique, the flanged-waveguide measurement geometry using either rectangular [27] or coaxial waveguides [28] and the dual-waveguide probe (DWP) geometry [29].

The PEC-backed DWP geometry yielded accurate [[mu].sub.r] results, but had difficulty in determining [[epsilon].sub.r] [29]. The inaccurate [[epsilon].sub.r] results were due to the fact that since the PEC-backed material under test (MUT) is typically electrically thin and the predominately transverse electric field in the MUT region is forced to zero at the PEC walls of the MUT/parallel-plate region, only a small interrogating electric field exists in the MUT to measure permittivity. The opposite is true for the magnetic field in the MUT region, thus physically explaining the experimental results. To specifically address the PEC-backed DWP's [[epsilon].sub.r]-measurement sensitivity, a free-space-backed (FS-backed) DWP geometry was recently analyzed [30]. Intuitively, the FS-backed geometry should be the complement of the PEC-backed geometry, i.e., permit a large electric field and small magnetic field into the MUT region. It logically follows that this arrangement should yield more accurate [[epsilon].sub.r] while yielding less accurate [[mu].sub.r] values. However, the FS-backed experimental results did not produce the expected improvement in [[epsilon].sub.r] due to insufficient coupling between the waveguide apertures, i.e., a substantial portion of the field was either reflected back from the aperture or radiated into the free-space region.

In this paper, a two-layer PEC-backed DWP geometry is proposed to improve the [[epsilon].sub.r] results over both existing PEC-backed and FS-backed geometries. In this geometry, the MUT is backed by a known material which is subsequently backed by a PEC (see Figure 1). This arrangement should allow a larger interrogating electric field into the MUT region (this of course depends on the electrical path length between the MUT and the PEC--[lambda]/4 being ideal for this purpose) while providing the necessary coupling between the probes. In the next section, the measurement structure depicted in Figure 1 is theoretically analyzed. Like the PEC-backed and FS-backed geometries, this is accomplished by replacing the waveguide apertures with equivalent magnetic currents in accordance with Love's equivalence theorem [31,32]. Enforcing the continuity of the transverse magnetic fields in the waveguide and parallel-plate regions at the waveguide apertures yields a set of coupled magnetic field integral equations (MFIEs). These MFIEs are then solved via the Method of Moments (MoM) [32,33] for the theoretical S-parameters, [S.sup.thy.sub.11] and [S.sup.thy.sub.21]. The permittivity and permeability of the MUT are then found by solving the nonlinear system

[absolute value of [S.sup.thy.sub.11] (f,d,l,[[epsilon].sub.k],[[mu].sub.k];[[epsilon].sub.r],[[mu].sub.r]) - [S.sup.meas.sub.11](f)] [??] [delta]

[absolute value of [S.sup.thy.sub.21] (f,d,l,[[epsilon].sub.k],[[mu].sub.k];[[epsilon].sub.r],[[mu].sub.r]) - [S.sup.meas.sub.21](f)] [??] [delta] (1)

numerically. Here, [delta] is a user-defined tolerance and f is the frequency; the other symbols are defined pictorially in Figure 1. In Section 3, the new method is experimentally validated. Measurements are made of two magnetic shielding materials, ECCOSORB[R] FGM-125 and FGM-40 [34], each backed by acrylic ([[epsilon].sub.k] [approximately equal to] 2.6, [[mu].sub.k] [approximately equal to] 1) and then a PEC. Last, this paper concludes with a summary of the work presented.

[FIGURE 1 OMITTED]

2. METHODOLOGY

2.1. System of Coupled MFIEs

The derivation of the theoretical coefficients [S.sup.thy.sub.11] and [S.sup.thy.sub.21] requires expressions for the transverse electric and magnetic fields in the waveguide/probe regions of Figure 1. In the probes, the fields may be expressed as a weighted summation of [TE.sup.z.sub.mn] and [TM.sup.z.sub.mn] rectangular waveguide modes [31]. In probe I, the transverse fields are comprised of an incident T[E.sup.z.sub.10] mode plus higher-order TE and TM reflected modes weighted by unknown and ultimately desired modal reflection coefficients [[GAMMA].sup.TE.sub.mn] and [[GAMMA].sup.TM.sub.mn]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

In probe II, the transverse fields are comprised of a sum of weighted transmitted modes, i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Here, an exp (j[omega]t) time dependence is assumed and suppressed; [e.sup.TE.sub.mn], [e.sup.TM.sub.mn], [e.sup.TE.sub.mn], and [e.sup.TM.sub.mn] are the transverse electric and magnetic field distributions; [[GAMMA].sup.TE.sub.mn], [[GAMMA].sup.TM.sub.mn], [[GAMMA].sup.TE.sub.mn], and [[GAMMA].sup.TM.sub.mn] are the transverse electric and magnetic field reflection and transmission coefficients; and [[gamma].sub.mn] = [([k.sup.2.sub.x] + [k.sup.2.sub.y] - [k.sup.2.sub.0]).sup.1/2] is the z-directed propagation constant. Due to the symmetry of the T[E.sup.z.sub.10] incident field and the DWP geometry, only [TE.sup.z.sub.mn] and [TM.sup.z.sub.mn] modes of odd m index are excited [29]. Of these, the [TE.sup.z.sub.1N / [TM.sup.z.sub.1n] modes are most significant [29,35].

The transverse magnetic field in the MUT region of Figure 1 is found by replacing the waveguide probe apertures with equivalent magnetic surface currents [M.sub.1] and [M.sub.2] via Love's equivalence principle [31,32]. The transverse magnetic field is computed using these currents through the electric vector potential F, i.e.,

[H.sup.pp.sub.t] = 1 / j[omega][[mu].sub.0][[mu].sub.r] ([k.sup.2.sub.MUT] + [[nabla].sub.t][nabla]*)F (4)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Note that [bar.G] is the dyadic Green's function for a magnetic-current-excited two-layer parallel-plate environment. In the DWP geometry analyzed in this paper, both the source and observation points are collocated in the MUT region of Figure 1. Thus, only that form (i.e., source in region 1, observer in region 1) of the Green's function is required. The derivation of the Green's function is provided in [36]. Only the relevant form needed for further theoretical analysis is provided here:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Note that [??] is the spectral-domain form of the Green's function; it is related to the spatial form [bar.G] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

In (6), [alpha] = x, y, [k.sub.n] = 2[pi]f [([[epsilon].sub.n][[mu].sub.n]).sup.1/2], and [p.sub.n] = [([[xi].sup.2] + [[eta].sup.2] - [k.sup.2.sub.n]).sup.1/2] where n = 1 (MUT region) or n = 2 (known backing region), respectively. Also in (6),

[N.sup.[+ or -].sub.cosh] = cosh[[p.sub.2](d - l)] {cosh[[p.sub.1](l - [absolute value of z - z'])][+ or -] cosh[[p.sub.1](l -(z + z')]}

[N.sup.[+ or -].sub.sinh] = sinh[[p.sub.2](d - l)] {sinh[[p.sub.1](l - [absolute value of z - z'])][+ or -]sinh [[p.sub.1](l -(z + z')]}

[D.sub.t] = [[epsilon].sub.2][p.sub.1] sinh([p.sub.1]l) cosh[[p.sub.2](d - l)] + [[epsilon].sub.1][p.sub.2] cosh([p.sub.1]l) sinh[[p.sub.2](d - l)]

[D.sub.n] = [[mu].sub.1][p.sub.2] sinh([p.sub.1]l) cosh[[p.sub.2](d - l)] + [[mu].sub.2][p.sub.1] cosh([p.sub.1]l) sinh[[p.sub.2](d - l)] (8)

The system of coupled MFIEs is derived by enforcing continuity of transverse magnetic fields at the waveguide probe apertures:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Note that the unknowns in the above system are [M.sub.1], [M.sub.2], [[GAMMA].sup.TE.sub.mn], [[GAMMA].sup.TM.sub.mn], [T.sup.TE.sub.mn], and [T.sup.TM.sub.mn].

2.2. MoM Solution

The MoM [32,33] is employed to solve the system of coupled MFIEs in (9). The first step in the MoM is to expand the unknown currents in a set of basis functions. Note that the magnetic current density is related to the electric field by

M = -[??] x E (10)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in this application. It thus makes physical sense to choose the transverse electric field distributions given in (2) and (3) as basis functions for [M.sub.1] and [M.sub.2], respectively:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Substitution of (11) into (9) and subsequent simplification yields a 4 x 4N system. In order to make the system full rank, the second step of the MoM is performed, namely, testing. Rectangular waveguide transverse magnetic field distributions are used as testing functions to take advantage of waveguide-mode orthogonality. The testing operation yields a 4N x 4N system of equations, which may be represented in matrix form as Ax = b where A is the MoM impedance matrix, b is a vector containing the contribution from the incident field, and x is a vector of the complex reflection and transmission coefficients.

2.3. Material Parameter Extraction

As with most material characterization methods, the notable exception being Nicolson, Ross, Weir (NRW) [37,38], a closed-form expression for [[epsilon].sub.r] and [[mu].sub.r] in terms of [S.sub.11] and [S.sub.21] does not exist. It thus becomes necessary to perform the inversion numerically to extract [[epsilon].sub.r] and [[mu].sub.r] from the theoretical models. To mitigate the effects of noise, S12 and S22 are included in the system of nonlinear equations, namely,

[absolute value of [S.sup.thy.sub.11] (f,d,l,[[epsilon].sub.k],[[mu].sub.k];[[epsilon].sub.r],[[mu].sub.r]) - [S.sup.meas.sub.11](f)] [less than or equal to] [delta]

[absolute value of [S.sup.thy.sub.21] (f,d,l,[[epsilon].sub.k],[[mu].sub.k];[[epsilon].sub.r],[[mu].sub.r]) - [S.sup.meas.sub.21](f)] [less than or equal to] [delta]

[absolute value of [S.sup.thy.sub.12] (f,d,l,[[epsilon].sub.k],[[mu].sub.k];[[epsilon].sub.r],[[mu].sub.r]) - [S.sup.meas.sub.12](f)] [less than or equal to] [delta]

[absolute value of [S.sup.thy.sub.22] (f,d,l,[[epsilon].sub.k],[[mu].sub.k];[[epsilon].sub.r],[[mu].sub.r]) - [S.sup.meas.sub.22](f)] [less than or equal to] [delta] (12)

Since the proposed system of equations is over determined a nonlinear least squares method, the Levenberg-Marquardt algorithm, is used to solve (12) to within a tolerance [delta] = [10.sup.-6] [39]. Note that [[GAMMA].sup.TE.sub.10] = [S.sup.thy.sub.11] = [S.sup.thy.sub.22] due to the symmetry of the measurement geometry and [T.sup.TE.sub.10] = [S.sup.thy.sub.21] = [S.sup.thy.sub.12] due to reciprocity.

3. EXPERIMENTAL VERIFICATION

The DWP consists of two precision X-band rectangular waveguides connected with screws to a 30.48 cm x 30.48 cm x 9.779 mm aluminum flange plate (see Figure 2). To ensure sufficient coupling when measuring lossy materials, the rectangular waveguides are machined so that only a 3.810 mm space exists between their apertures. The measurement apparatus is calibrated using the Thru-Reflect-Line calibration method (TRL) [40] using specially machined line and short standards. The TRL calibration places the forward and reverse phase-reference planes at the probe I and probe II rectangular waveguide apertures. To shift these calibration planes to the flange/material interface, an ideal phase delay, equal to twice the T[E.sup.z.sub.10] z-directed propagation constant [[gamma].sub.10] times the aluminum flange plate thickness, is applied to the raw measured S-parameters, i.e.,

[S.sup.meas.sub.11] = [S.sup.raw.sub.11] exp(2[[gamma].sub.10][DELTA])

[S.sup.meas.sub.21] = [S.sup.raw.sub.21] exp(2[[gamma].sub.10][DELTA])

[S.sup.meas.sub.12] = [S.sup.raw.sub.12] exp(2[[gamma].sub.10][DELTA])

[S.sup.meas.sub.22] = [S.sup.raw.sub.22] exp(2[[gamma].sub.10][DELTA]) (13)

where [DELTA] is the flange plate thickness. To remove any possible reflections from the flange plate edges, the measured S-parameters are time gated as discussed in [41]. Also to minimize the effects of small air gaps which might exist between the flange plate and the MUT, approximately 18 kg of weight is applied to the top of the aluminum flange near where the rectangular waveguides attach to the plate during S-parameter measurements.

[FIGURE 2 OMITTED]

Material measurements at X-band (8.2 GHz to 12.4 GHz) of 3.175 mm thick ECCOSORB[R] FGM-125 and 1.016 mm thick ECCOSORB[R] FGM-40 [34] were made using an Agilent Technologies E8362B vector network analyzer (VNA) [42]. A 30.48 cm x 30.48 cm x 5.5 mm sample of acrylic ([[epsilon].sub.k] [approximately equal to] 2.6, [[mu].sub.k] [approximately equal to] 1) was used as the known backing material in all measurements utilizing the two-layer DWP geometry. Acrylic is a rigid, low-loss, low-dispersion material making it very well suited to serve as a backing material. The [[epsilon].sub.k] and [[mu].sub.k] values for acrylic used in the extraction algorithm were determined using the standard NRW technique.

3.1. FGM-125 Results

The measured real and imaginary parts of both the permittivity [[epsilon].sub.r] [Figures 3(a) and 3(b)] and permeability [[mu].sub.r] [Figures 3(c) and 3(d)] of FGM-125 are shown in Figure 3. In the (a) and (c) subfigures, the PEC-backed (circle traces) and two-layer (square traces) DWP results are plotted; in the (b) and (d) subfigures, the two-layer (square traces) and FS-backed (triangle traces) DWP results are plotted. Each DWP geometry result has two traces associated with it--1 and 5 [TE.sup.z.sub.1n] / [TM.sup.z.sub.1n] higher-order modes traces denoted by different colors and line styles. For reference, the results using the industry standard NRW destructive characterization technique are also plotted. Complex permittivity and permeability for FGM-125 were also obtained using 10 higher-order modes. These traces are not included so as not to clutter Figure 3. Root-mean-square errors (RMSEs) of the PEC-backed, FS-backed, and two-layer DWP results (1, 5, and 10 higher-order modes) assuming the NRW results are the true values are reported in Table 1. Note that no real improvement in [[epsilon].sub.r] or [[mu].sub.r] accuracy results from including more than 5 higher-order modes implying solution convergence.

Certainly of note from the results presented in Figure 3 and Table 1 is that the [[epsilon].sub.r] results using the two-layer DWP geometry are significantly better than either the PEC-backed or FS-backed geometries, although the errors do become comparable as mode number increases for the PEC-backed results. The two-layer DWP [[epsilon].sub.r] improvement is physically expected considering that for the PEC-backed measurement, due to the distribution of the T[E.sup.z.sub.10] incident field and the relative thinness of the MUT, only a small interrogating electric field exists in the MUT region to probe the MUT's electrical properties. This is contrasted with the two-layer DWP which moves the MUT away from the PEC backing, thus changing the nature of the fields in the MUT region. Since the MUT is no longer backed by PEC, the interrogating electric field in the MUT region is much stronger than in the PEC-backed case, thus providing a more accurate and reliable [[epsilon].sub.r] measurement. In the case of [[mu].sub.r], the results for the two-layer DWP are more error prone than the PEC-backed geometry. This result is also physically expected considering the behavior of the magnetic field in the two DWP geometries. For the PEC-backed measurement, the magnetic field in the MUT region can be expected to be large considering the proximity to the PEC boundaries. Thus, one would expect that the [[mu].sub.r] measurement would be accurate and reliable. Contrast this once again with the two-layer DWP geometry; the MUT is now located away from the PEC. While this action increases the strength of the electric field in the MUT region (as previously discussed), this increase in electric field strength comes at the expense of the magnetic field. This explains the less accurate [[mu].sub.r] results for the two-layer DWP.

[FIGURE 3 OMITTED]

Applying the physical reasoning of the previous paragraph, one would expect that the two-layer DWP would perform about the same (perhaps a little worse) compared to the FS-backed geometry for [[epsilon].sub.r] and better for [[mu].sub.r]. However, as described in detail in [30], the FS-backed DWP suffers from poor probe coupling (highly dependent on the MUT), i.e., very little energy is transmitted from probe I, through the MUT, to probe II. This makes full electromagnetic characterization (determining [[epsilon].sub.r] and [[mu].sub.r] unambiguously) very difficult considering that only the reflection measurements are viable. Note that it was this disappointing result which served as the impetus for the two-layer DWP geometry which confines the interrogating field between parallel, PEC plates thus significantly improving probe coupling, i.e., transmission.

3.2. FGM-40 Results

The measured real and imaginary parts of [[epsilon].sub.r] [Figure 4(a)] and [[mu].sub.r] [Figure 4(b)] for the thinner and more heavily loaded FGM-40 are shown in Figure 4. In the figures, both the two-layer (square traces) and FS-backed (triangle traces) DWP results are plotted. Note that the PEC-backed results are not presented because a solution to (12) is not found. FGM-40 is too electrically thin to numerically calculate the theoretical S-parameters using the PEC-backed DWP geometry. As is the case for the FGM-125 results, each DWP geometry has two traces associated with it, namely, 1 and 5 [TE.sup.z.sub.1n] / [TM.sup.z.sub.1n] higher-order modes traces demarcated by different colors and line styles. The NRW results for FGM-40 are included on the figures as a reference. Table 2 reports the RMSEs of the FS-backed and two-layer DWP results assuming that the NRW [[epsilon].sub.r] and [[mu].sub.r] are the true values.

[FIGURE 4 OMITTED]

Inspection of the results presented in Figure 4 and Table 2 reveals that the two-layer DWP geometry improves upon the FS-backed geometry in every regard. As is the case for FGM-125 discussed in Section 3.1, the FS-backed DWP still experiences poor transmission; however, in this case, [S.sub.21] is sufficiently above the VNA noise floor to yield results for [[epsilon].sub.r] and [[mu].sub.r]. However, since [S.sub.21] is still very weak, the results are more error prone than the two-layer DWP geometry.

3.3. Combined Results

The last results presented in this paper are the combined DWP geometry results, i.e., the PEC-backed geometry combined with the two-layer and FS-backed DWP geometries. In these results,

[absolute value of [S.sup.thy.sub.PEC] ([[epsilon].sub.r],[[mu].sub.r]) - [S.sup.meas.sub.PEC]] [less than or equal to] [delta]

[absolute value of [S.sup.thy.sub.TL] ([[epsilon].sub.r],[[mu].sub.r]) - [S.sup.meas.sub.TL]] [less than or equal to] [delta]

[absolute value of [S.sup.thy.sub.PEC] ([[epsilon].sub.r],[[mu].sub.r]) - [S.sup.meas.sub.PEC]] [less than or equal to] [delta]

[absolute value of [S.sup.thy.sub.FS] ([[epsilon].sub.r],[[mu].sub.r]) - [S.sup.meas.sub.FS]] [less than or equal to] [delta] (15)

are solved numerically (eight equations for two unknowns). Here, S = [[S.sub.11] [S.sub.21] [S.sub.12] [S.sub.22]], TL represents the two-layer DWP geometry, and FS represents the FS-backed DWP geometry. The impetus for combining the different DWP geometries in this manner arose from the analysis performed in [30]. Figure 5 shows the combined PEC-backed and two-layer DWP geometries (square traces) and the combined PEC-backed and FS-backed DWP geometries (triangle traces) [[epsilon].sub.r] [Figure 5(a)] and [[mu].sub.r] [Figure 5(b)] FGM-125 results for 1 and 5 T[E.sup.z.sub.1n]/T[M.sup.z.sub.1n] higher-order modes (demarcated by different colors and line styles). Table 3 compares the RMSEs (FGM-125 NRW [[epsilon].sub.r] and [[mu].sub.r] results assumed to be the true values) for the combined PEC-backed and FS-backed DWP geometries with those of the combined PEC-backed and two-layer DWP geometries.

[FIGURE 5 OMITTED]

Comparison of the combined PEC-backed and two-layer DWP errors listed in Table 3 with those listed for the two-layer DWP in Table 1 implies that little is gained by using measurements from both PEC-backed and two-layer DWP geometries. This is in stark contrast to the combined PEC-backed and FS-backed [[epsilon].sub.r] and [[mu].sub.r] errors (listed in Table 3) which are much better than those reported using the FS-backed DWP alone (see Table 1 and [30]). Note that these results imply that while the PEC-backed and two-layer DWP measurements are different, no new information is gained (i.e., the measurements are not completely independent) and thus do not yield more accurate [[epsilon].sub.r] and [[mu].sub.r] results when combined in the manner outlined in (14). The opposite is true for the PEC-backed and FS-backed DWP measurements. These two measurements (analogous to the independent combination of PEC-backed and matched-termination-backed [S.sub.11] waveguide measurements) provide new information thus yielding more accurate [[epsilon].sub.r] and [[mu].sub.r] values when combined in the manner specified by (15). Note that combined measurements of FGM-40 were also made. However, like in [30], these results mirror the two-layer and FS-backed DWP [[epsilon].sub.r] and [[mu].sub.r] values reported in Figures 3 and 4 and Tables 1 and 2 because of the aforementioned difficulty in calculating theoretical S-parameters using the PEC-backed structure.

4. CONCLUSION

A nondestructive technique for determining the complex permittivity and permeability of a planar material using a two-layer dual-waveguide probe measurement geometry was presented. The impetus for this new dual-waveguide probe geometry was two fold. The first was to address the [[epsilon].sub.r]-measurement sensitivity experienced by the PEC-backed geometry [29] by displacing the MUT from the PEC by a known material thereby permitting a larger electric field interrogation of the MUT region. The second was to address the weak field coupling between waveguide probes experienced by the FS-backed geometry [30] by containing the measurement in a parallel-plate environment (see Figure 1). The experimental results clearly demonstrated the utility of this approach. Significant improvements in permittivity were observed using the two-layer geometry compared to both the PEC-backed and FS-backed geometries.

The views expressed in this paper are those of the authors and do not reflect the official policy or position of the U.S. Air Force, the Department of Defense, or the U.S. Government.

Received 11 November 2011, Accepted 7 December 2011, Scheduled 14 December 2011

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M. D. Seal, M. W. Hyde *, and M. J. Havrilla

Air Force Institute of Technology, 2950 Hobson Way, Wright-Patterson AFB, OH 45433, USA

* Corresponding author: Milo Wilt Hyde (milo.hyde@afit.edu).
Table 1. PEC-Backed, FS-Backed, and
Two-Layer DWP RM FGM-125.

                               PEC-Backed DWP

                         1 Mode   5 Modes   10 Modes

Re [[[epsilon].sub.r]]   2.8407   1.0486     0.8491
Im [[[epsilon].sub.r]]    2.33    0.6416     0.5974
Re [[[micro].sub.r]]     0.135    0.0253     0.0246
Im [[[micro].sub.r]]     0.1957   0.0655     0.0531

                               FS-Backed DWP

                         1 Mode   5 Modes   10 Modes

Re [[[epsilon].sub.r]]   1.1126   0.8152     0.7768
Im [[[epsilon].sub.r]]   3.1773   2.1899     2.0627
Re [[[micro].sub.r]]     0.3298   0.1833     0.1683
Im [[[micro].sub.r]]     0.1611   0.1493     0.1414

                               Two-Layer DWP

                         1 Mode   5 Modes   10 Modes

Re [[[epsilon].sub.r]]   0.6508   0.4454     0.4225
Im [[[epsilon].sub.r]]   0.4095   0.2279     0.2043
Re [[[micro].sub.r]]     0.0549   0.0574     0.0542
Im [[[micro].sub.r]]     0.1018   0.0659     0.0575

Table 2. FS-Backed and Two-Layer DWP
RMSEs for FGM-40.

                               FS-Backed DWP

                         1 Mode   5 Modes   10 Modes

Re [[[epsilon].sub.r]]   2.6593   2.5588    2.4356
Im [[[epsilon].sub.r]]   1.7189   1.5725    1.6265
Re [[[micro].sub.r]]     0.1799   0.1213    0.1117
Im [[[micro].sub.r]]     0.1546   0.1349    0.1381

                               Two-Layer DWP

                         1 Mode   5 Modes   10 Modes

Re [[[epsilon].sub.r]]   1.3815   1.1886    1.106
Im [[[epsilon].sub.r]]   0.8218   0.4611    0.4609
Re [[[micro].sub.r]]     0.0731   0.0524    0.0495
Im [[[micro].sub.r]]     0.0683   0.0973    0.0937

Table 3. Combined PEC-Backed and FS-Backed
DWP and Combined PEC-Backed and Two-Layer
DWP RMSEs for FGM-125.

                         PEC-Backed & FS-Backed DWP

                         1 Mode   5 Modes   10 Modes

Re [[[epsilon].sub.r]]   0.2302   0.1308    0.0774
Im [[[epsilon].sub.r]]   0.3528   0.1893    0.1734
Re [[[micro].sub.r]]     0.0221   0.0236    0.0189
Im [[[micro].sub.r]]     0.0514   0.0092    0.0096

                         PEC-Backed & Two-Layer DWP

                         1 Mode   5 Modes   10 Modes

Re [[[epsilon].sub.r]]   0.2563   0.232     0.1748
Im [[[epsilon].sub.r]]   0.3388   0.1952    0.1763
Re [[[micro].sub.r]]     0.0256   0.0305    0.0259
Im [[[micro].sub.r]]     0.0477   0.011     0.0095
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Author:Seal, M.D.; Hyde, M.W.; Havrilla, M.J.
Publication:Progress In Electromagnetics Research
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Date:Jan 1, 2012
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